Broad quantum state tomography model based on adaptive feature extraction

doi:10.11772/j.issn.1001-9081.2023121725

Abstract

Abstract:

Aiming at the problem of exponential growth of data dimension faced by Quantum State Tomography （QST）， a Broad QST model based on Adaptive Feature Extraction （AFE_BQST） was proposed. Firstly， an adaptive feature extraction strategy was introduced to avoid the uncertainty of mapping feature nodes caused by random generation of weights. Secondly， Broad Learning System （BLS） was used to map the input data to a more appropriate feature space in a non-iterative way for feature extraction of large-capacity data. Finally， experiments were executed in the cases of low- and high-dimensional quantum state data to compare AFE_BQST with Broad QST （BQST）， Deep neural network QST （D_QST）， Convolutional neural network QST （C_QST） and U-shaped network QST （U_QST） models by using two indicators of average fidelity and running time. Experimental results show that in the case of small samples with low-dimensional quantum state， compared with the sub-optimal baseline model BQST， AFE_BQST improves the fidelity by 0.045 percentage points with the similar running time； in the case of large samples with high-dimensional quantum state， compared with the sub-optimal baseline model D_QST， AFE_BQST improves the fidelity by 0.175 percentage points with the running time reduced by 99%. The above results prove that AFE_BQST is able to extract quantum state data features adaptively and reconstruct quantum state data accurately and efficiently.

Key words: Quantum State Tomography (QST), Broad Learning System (BLS), adaptive feature extraction, deep learning, fidelity

摘要：

针对量子态层析（QST）面临的数据维度指数级增长的难题，提出一种基于特征自适应提取的宽度QST模型（AFE_BQST）。首先，引入特征自适应提取策略，从而避免权重的随机性生成导致的映射特征节点不确定的现象；其次，采用宽度学习系统（BLS），以一种非迭代的方式将输入的数据映射到更合适的特征空间，提取大容量数据特征；最后，在低维和高维量子态数据的情况下进行实验，采用平均保真度和运行时间这2个性能指标，将AFE_BQST与宽度QST（BQST）、基于深度神经网络的QST（D_QST）、基于卷积神经网络的QST（C_QST）和基于U形神经网络的QST（U_QST）模型对比。实验结果表明，在低维量子态小样本情况下，AFE_BQST与次优基线模型BQST相比，提升了0.045个百分点的平均保真度，且运行时间相当；在高维量子态大样本情况下，AFE_BQST与次优基线模型D_QST相比，提升了0.175个百分点的平均保真度，且减少了99%的运行时间。以上结果验证了AFE_BQST具有自适应提取量子态数据特征以及准确且高效地重构量子态数据的能力。

关键词: 量子态层析, 宽度学习系统, 特征自适应提取, 深度学习, 保真度

CLC Number:

TP391.9

Wenjie YAN, Dongyue DANG. Broad quantum state tomography model based on adaptive feature extraction[J]. Journal of Computer Applications, 2024, 44(12): 3861-3866.

闫文杰, 党东月. 基于特征自适应提取的宽度量子态层析模型[J]. 《计算机应用》唯一官方网站, 2024, 44(12): 3861-3866.

Figures/Tables 8

Tab. 1 Explanation of relevant mathematical symbols

变量	含义	备注
$ρ$	量子密度矩阵	矩阵
$ψ$	量子态	矢量
$H$	水平极化状态	矢量
$V$	垂直极化状态	矢量
$u$	酉矩阵	矩阵
$I$	单位矩阵	矩阵
$L$	Cholesky分解所得下三角矩阵	矩阵
$T$	量子态重构参数	矢量
$M$	测量算子	矩阵
$σ$	泡利算子	矩阵
$m$	测量数据	矢量
$X$	输入数据	矩阵
$Y$	输出数据	矩阵
$Z p$	p组映射特征节点	矩阵
$E q$	q组增强节点	矩阵
$W$	输出权重矩阵	矩阵
$A$	BLS模型总输入	矩阵

Tab. 1 Explanation of relevant mathematical symbols

变量	含义	备注
$ρ$	量子密度矩阵	矩阵
$ψ$	量子态	矢量
$H$	水平极化状态	矢量
$V$	垂直极化状态	矢量
$u$	酉矩阵	矩阵
$I$	单位矩阵	矩阵
$L$	Cholesky分解所得下三角矩阵	矩阵
$T$	量子态重构参数	矢量
$M$	测量算子	矩阵
$σ$	泡利算子	矩阵
$m$	测量数据	矢量
$X$	输入数据	矩阵
$Y$	输出数据	矩阵
$Z p$	p组映射特征节点	矩阵
$E q$	q组增强节点	矩阵
$W$	输出权重矩阵	矩阵
$A$	BLS模型总输入	矩阵

Fig. 1 Structure of BLS

Fig. 2 Structure of AFE_BQST

Tab. 2 Relevant parameters of model required for experiments

量子数据	模型	Feature_layer	Node	Epoch	Batch	Learner	Learning_rate
2量子比特	AFE_BQST	36-36-36	108-500	10	32	Adam	0.001 0
	BQST		108-500
	D_QST			50	32	Adagrad	0.000 5
	C_QST			50	4	Adagrad	0.008 0
	U_QST			50	4	Adagrad	0.001 0
4量子比特	AFE_BQST	800-400-200	1 400-800	10	32	Adam	0.001 0
	BQST		1 400-800
	D_QST			50	32	Adagrad	0.000 6
	C_QST			50	4	Adagrad	0.006 0
	U_QST			50	4	Adagrad	0.001 0

Fig.3 Average fidelities of 2 qubits and 4 qubits at different learning rates

Fig. 4 Experimental results of 2 qubits under ideal data

Fig. 5 Experimental results of 4 qubits under ideal data

Tab. 3 Experimental results of different sample sizes under 2 qubits and 4 qubits

量子数据/qubit	数据样本数	模型	Avr_F/%	运行时间/s	量子数据/qubit	数据样本数	模型	Avr_F/%	运行时间/s
2	200	D_QST	92.827	18.825	4	800	D_QST	94.762	171.531
		C_QST	92.237	31.592			C_QST	88.498	166.740
		U_QST	94.841	52.555			U_QST	85.507	478.890
		BQST	96.576	1.076			BQST	95.921	5.237
		AFE_BQST	96.621	1.465			AFE_BQST	94.349	7.554
	400	D_QST	95.914	37.658		1 000	D_QST	95.281	232.766
		C_QST	94.580	63.128			C_QST	90.761	213.430
		U_QST	96.830	106.193			U_QST	88.133	645.320
		BQST	92.539	1.480			BQST	96.476	6.134
		AFE_BQST	97.240	2.097			AFE_BQST	94.776	8.593
	1 000	D_QST	97.752	94.290		6 000	D_QST	98.660	1 495.239
		C_QST	97.797	155.443			C_QST	96.378	1 263.060
		U_QST	98.609	256.514			U_QST	98.237	4 155.660
		BQST	97.749	2.679			BQST	94.424	20.681
		AFE_BQST	97.911	4.217			AFE_BQST	99.119	36.396
	2 000	D_QST	98.276	188.897		8 000	D_QST	98.989	1 585.457
		C_QST	98.709	312.258			C_QST	96.768	1 662.183
		U_QST	99.208	554.446			U_QST	98.704	5 560.150
		BQST	98.842	4.785			BQST	97.632	27.318
		AFE_BQST	98.089	8.359			AFE_BQST	99.233	47.251
	4 000	D_QST	99.208	379.409		10 000	D_QST	99.162	1 763.861
		C_QST	99.095	629.247			C_QST	96.925	2 056.470
		U_QST	99.457	1 099.216			U_QST	98.903	6 770.020
		BQST	99.212	8.359			BQST	98.531	34.885
		AFE_BQST	98.335	15.755			AFE_BQST	99.337	59.621

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